Weak solutions for anisotropic nonlinear elliptic equations with variable exponents
We study the anisotropic boundary-value problem $$displaylines{ -sum^{N}_{i=1}frac{partial}{partial x_{i}}a_{i}(x,frac{partial}{partial x_{i}}u)=f quad hbox{in } Omega, cr u=0 quadhbox{on }partial Omega, }$$ where $Omega$ is a smooth bounded domain in $mathbb{R}^{N}$ $(Ngeq 3)$. We obtain th...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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Texas State University
2009-11-01
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| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2009/144/abstr.html |
| Summary: | We study the anisotropic boundary-value problem $$displaylines{ -sum^{N}_{i=1}frac{partial}{partial x_{i}}a_{i}(x,frac{partial}{partial x_{i}}u)=f quad hbox{in } Omega, cr u=0 quadhbox{on }partial Omega, }$$ where $Omega$ is a smooth bounded domain in $mathbb{R}^{N}$ $(Ngeq 3)$. We obtain the existence and uniqueness of a weak energy solution for $fin L^{infty}(Omega)$, and the existence of weak energy solution for general data $f$ dependent on $u$. |
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| ISSN: | 1072-6691 |